On the mod-$\mathrm{\ell }$ homology of the classifying space for commutativity

We study the mod-$\mathrm{\ell }$ homotopy type of classifying spaces for commutativity, $B(\mathbb{Z},G)$, at a prime $\mathrm{\ell }$. We show that the mod-$\mathrm{\ell }$ homology of $B(\mathbb{Z},G)$ depends on the mod-$\mathrm{\ell }$ homotopy type of $BG$ when $G$ is a compact connected Lie group, in the sense that a mod-$\mathrm{\ell }$ homology isomorphism $BG\to BH$ for such groups induces a mod-$\mathrm{\ell }$ homology isomorphism $B(\mathbb{Z},G)\to B(\mathbb{Z},H)$. In order to prove this result, we study a presentation of $B(\mathbb{Z},G)$ as a homotopy colimit over a topological poset of closed abelian subgroups, expanding on an idea of Adem and G'omez. We also study the relationship between the mod-$\mathrm{\ell }$ type of a Lie group $G(\mathbb{C})$ and the locally finite group $G({\bar{\mathbb{F}}}_p)$ where $G$ is a Chevalley group. We see that the na"ive analogue for $B(\mathbb{Z},G)$ of the celebrated Friedlander–Mislin result cannot hold, but we show that it does hold after taking the homotopy quotient of a $G$ action on $B(\mathbb{Z},G)$.